Aircraft Anti-lock Brake System (ABS) with Fuzzy Logic Control

Copyright © IFAC Automatic Control in Aerospace. l'a1o Alto. CaWomia. USA. 1994

Aircraft Anti-lock Brake System (ABS) with Fuzzy Logic Control H. Chris Tseng* and Charlie W. Chi* *

Int~llIg.:nt

Control Labor:IlOI\" EIe.:trIcal Engll1cerIng I)cparlm.:nL Santa Clara Uni\'erslt\, Santa Clnra CA

')5035, USA

Ahslract. Fuzz\' logIc IS used to d..:al "'ith nonl1l1ear ABS that contalIls lIlcomplete or uncerlnln characterislics, A nonllIlcar lI1tcll)olation dcslgn from gl\,en look-up tables "'nh fuzzy logic IS proposed The resulting contlIluous 1I1tcllxllatcd nonl111car ch:lractCrIstlc,; are uscd as id..:ntlfi.:d parameters 111 the system design The heUristic ruleh:ls~d fuzzy logIC controllcr IS sho"n to achi..:\c ma:\l1l1Um dragg1l1g force 111 ,;111lUlatlon Key \Vonls. AntI-lock hr.lk.: s\stcm, fuzz\' logl': control. nonl111car control: ;l1rcraft

I.

INTRODUCTION

technique for nonlinear 111terpoiatlon uSlllg such arc introduced The main problem statement and the rule based fuzzy logIC ABS deSIgn arc introduced in the subsequent section Wc cnd thIS paper in the conclusion section by prO\iding dIrections for futllre research 111 aerospace control and suggesting areas where integration of modern control theory \\'lth intelligent technIques can provide a powerful tool for the de,elopment of future intelligent controllers

Anti-lock brake system (ABS) has been widely used in both automobiles and aircraft. Due to the uncertain nature of the road conditions and the complicated nonlinear dynamics of wheels coupled "ith the aircraft system. ABS design Imposcs a challenging ISSUC for control and aerospace engll1eers Recentl~ a study released by the United States government re"ealed that the e:\lsting ABS deSigns III more than 10 millIon cars 111 the US did not delIver their promise and called for more research work (San Jose Mercury Ne\\s, 1994) Numerolls companies. including McDonnell Douglas and Ford Motor. are actively investigating this subJect Fuzzy logic has succeeded in many control problems where tradItional approaches have failed because fuzzy logIC comes to gnps with the per"asi,'e Imprecision of the real world and offers a methodology for e:\ploiting the tolerance for imprecision to achieye tractability. imprO\'ed perfonnance. and power solution cost The fuzzy logic based self tuning PI controller by Tseng and Hmlllg (1993) is onc of the designs that demonstrates its superior performance over the conventional controller for the yague and noisv systems,

2.

FUZZY LOGIC CONTROLLER

Ever since Zadeh Illtroduced the concept of fuzzy set theory in 1965 (Zadeh, 1965), it has found many applications in a variety of fields, Onc of the most llnp0l1ant and successful applications of fuzzy set theory IS the subject of fuzzy logic control (Mamdani and Assilian. 1975)

I n tillS phi losophy \\e Illtegratc fuzzy logIC WIth the A BS deSIgn and present our work as follows In section 2. a bricfrevie\\ of fuzzy logic and a

387

equi\'alent linguistic parameters such as (negorive, zero. positive} with associated membership function values . Various rules in the knowledge base and the decision making logic nre invoked and contribute their control actions \,ith different degrees of emphasis depending on the respective membership values. A typical fuzzy rule for the henter control problem can be .

2.1 Basis of fuzzy logic

Positive

Negative

If Error IS Negallve Big and Change IS Zero

x

Zero 12

Then Change Fig .1. Fuzzy membership functions

~

!

I

I

I

Conrrol,s Poslnve BIg.

The centrOld defuzzilication is a common methodology and consists of functions of the form

x

x

'Z.OIl L l ..;

Fig . 2 . Common membership functions

J = 1

/j,1I

l

FlIzzili~r

Illrl?l'~Il(,~ Engin~

,=1

I'LAl\'T

D

\.

(x) I

_' _= _1_ __

m

I

Error

The final stage 111 the fuzzy logic controller architecture aggregates all the inferred fuzzy control action nnd produces a nonfuzzy numerIcal control through the defuzzificatlon module

~ (x)

(x)

!

/11

In

(21 )

"

I

= I

\\here 11-1 (X,) is thc assoclnted memberslllp \nlue of the ith II1pUt for the/th heUristic ll.Ile , z, is the equi\alent numerical control action to be fired upon for thc./th mle . and /j, 11 is the resulting aggregnte control change .

d

2.2 Nonlinear interpolation using fuzzy logic

Fig . 3. General architecture of a fuzzy logic controller system

Often only a limited number of data points IS available for certain nonlinear characteristics. In the aircraft anti-skid model one has the tables of tire deflection 0" \s. \'ertical load FI' and coefficient offriction R.-IT vs . slipping velOCIty r's as shown (see Table I nnd Tablc 2)

Unlike a CriSp seL fuzzy sets allo\\s partial membership . For example. a sensor reading of 1.2 may have membership \,lIues of 05 and 0.2 for bemg positi\'e and zero. respecti\'ely The partial membership for the sensor reading x is characterized by the fuzzy membership functions (see Fig. I) and can be expressed as 11 (x) = 0.2 and 11 (x) = O.S. Other l"" )1O .'i lfn'f! l"" I1f!S,UI\'(! popular membership functions may take onc of the forms shown in Figure 2 (sce FIg. 2) The basic architecture of fuzzy logic controller is illustrated (sec Fig. 3) The numerical output data from the output sensor of the process under control is codified through the fuzzilier into the

Il

TABLE 1.

f1 F/ , (/bs)

0 10,000 30,000 110,000

388

On (feel)

00 0104 0.213 0.667

TABLE 2.

f2 I', ((t!sec )

The defuzzification (2 2) is used to determine the tire deflection 0{J due to a vertical load Fr ' as follows :

~R .'T

0.0 1.0 0.94 0.50

000 13.43 25.45 60 .31

l

Fuzzy logic \\111 bc uscd to Intcrpolatc thc nonlincaritles This techlllque IS dCrl\'cd from thc conccpt of a fuzzy graph (Zadch . 1971) Basically, cach table entry is con\crtcd into a fuzzy rule . Then apply thc fuzzification with thc appropriatc mcmbcrship functions to thc fuzzy rule base to obtain the nonlincar II1tcrpolation curycs.

\\'here ~ , (F , .) is the associated membership yalue ofthe.lth heuristic rule, and 80 is the equivalent numerical data associated\vith the jth table entry of the tire deflection

ot -·

07 I'

J

101..: .1) 66 7}

Os. 0'.

li .

"

03 . 02

06- ---2

- -

7, - - - - . : -

F,.(klhs)

Fig . 5. Non-linear interpolation of Table 1

o 7~ " o 6 i ~.

I :

~(F ,. )

./

06:

PH

1'\ 1

(22)

, = I

Sincc thcre arc only fOllr numcncal data points a\ailablc for Tablc I. assIgn cach tirc dcflection data a fuzzy mcmbershlp dcnotcd b~ ZE (Zero). PS (Posltiyc Small), PM (Positi\c Medium), and PB (Positive Big) Thc shape of ca ch membership function is the Gaussian type mcmbership. Table I is di\'ided into four fuzzy membership regions where the mcan of each membership function corrcsponds to the gl\'cn vcrticalload for F , . (sce FIg 4)

ZE I'S

.

The fuzzy logic has the "generalization" feature that is capable of pro\'iding the associated non linear characteristics beyond the look-up table entries (see Fig. 5). Similarly, a set offuzzy "I f- Then" ntles can be generated from the given data paIrs tn Table 2 as follows

o s ~ ,.

I·

041 : . I . o 3[ : o 2[ 0'1 Ob'"-~---,---.t-· - .- 6"

- ;1) ' -----,'2'

F, . (klb .\)

I. If I 's isZEThen ~RAT isZE: 2. If I's isPSThen ~RAT isPB: 3. If I 's is PM Then ~ RA T is PM:

Fig. 4 . Fuzzy membership associated with Table 1

4. If I's isPBThen

The ncxt task hcre is to COI1\'crt thc table entries into a set of fuzzy rules from the gi\en data pairs. For each pair of data, one correspondtng "IfThen" rule is constructed . Four such ntles are constructed in this way

1. If 2. If 3. If 4. If

~RAT

is PS,

where I's IS the sensor measurement of the slipping velocity in ft./sec Figure 6 shows the Gaussian type fuzzy membership assigned to each linguistic description of the associating coefficient ~R.-;T (sec Fig. 6) Figure 7 shows the fuzzy logic-based interpolation contains the four entries from Table 2 and therefore is a good approximation of the associated non linear characterIstics (sce Fig. 7)

Fr' is ZE Then 00 is ZE : Fr' is PS Then 00 is PS : Fr ' is PM Then 00 is PM. Fr ' is PB Then 8{J is PB

389

' -'AC l E

PS

I'~I

o:r--~----- -

'f :i

16.6Ib·ft · se

'

\

FD

o 4~

= drag force

i

o 3[ o 2l

o

brake torque

/= moment of inertia of the wheel =

i

o o 6~

0Si

TB (I) =

II

0.1I \" 11 ( / ·s)

PH

----- )

= aircraft speed = 203 ft.lsec

Ro = radius of free tire = 185 ft

~1,' ,----.clorr-""'20.~-. '40--"~-ou--rO- --~0

(sec Table I)

/ ·s Vr I se, )

F I , = vertical load on tire

Fig. 6. Fuzzy membership associated with Table

, 's = slipping yelocity

2 RR = rolling radiUS

The sllnplified ABS prototype deSign will focus on the wheel speed control \\'hereas the aircraft speed is assumed to be constant The objectiye of this simplified ABS control problem is to maintain a highest possible IlR A T which occurs when the slipping speed " s = 13.43 ft/sec as II1dicated in Table 2. The \ertical load F I , = 25, (lOOlb:;; IS assumed and use equations (3.3)-(34) to ~Iculate the corresponding desired wheel speed (11 of 106.6 rad/sec that achieves maximum brakll1g drag force . The simplified ABS problem is then equl\alent to that ofa constant set-point trackll1g problem

o 6~

o '! I o 6f

Il rUT o s[

0.1r

031 02

I

1 .

o 1[ . I

°O-----W·- ~IJ·- ""30· - " -40 - - '-'>0-

I - 00- - - ' 0 -60

/ 's (/I I se, I

Fig. 7. Non-linear interpolation of Table 2

3.

ABS DYNAMICS AND FLC

The difTerence bet\\een the \\'heel speed III (I) and that of the deSired speed III is defined as

The wheel dynamics and the associated set-point control problem fOnlllllation are presented in this section . The \\heel speed dynamics 111 an aircraft subJect to brakll1g are as follo\\5

(11

=

L'

(!) =

the rate of change of

I II -

L'

III

(I )

(3 :\)

IS gl\en by

(3 I)

A typical rule for the control TB to achieve the set-point tracki!1g objectiYe . ie. lim Cl) (t) = fl) • can be :

(32) (3 .3)

I-+~

If e is something and (' IS something

IS

something Then

t>TB

(34) The input control TB IS bell1g updated as where

III (I)

= wheel speed

(3 7)

390

The fuzzy rules are summari7.ed in the following table . TABLE 3 . Fuzzy rule base

e

Rule no .

I:!

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

NB NM

!

",11

l\:~Il\:SlE

-- - -

0 6' ~ (e )

NB NM

0.4'

0 .2,

NS NB

~

'\13

..

~

50

'\\1 :-: S Z r. PS 1'\ 1

I'll

~ ( ,; )

PM lE lE

PS NS

'

0

1---- ---·-- - ----.--

NM NM

1\TB

~

0 · 50

NM NS PB PM PS lE PM

NB

I'll

0 .8:

PB PM PS

NM NS lE PS PB NS

PS PM I

04 0 .2

0 -- -

For this particular problem. sc\en fuzzy membershIp functions \yere used shO\\I1 in Figure 8, which are similar to those used by Lee (1990) for fuzzy logic control (see Fig 8) Table 4 summarizes the membership function description for (' , i!. and ,.., Tfl TABLE 4.

-

t. TIl

lE lE lE PB PM PS lE lE lE

NS lE lE lE PB PM PS lE lE lE lE NB NS PB PS NS PS

i

1 ,-

· 10

o

·5

'\"

'\5 lE PS

5

P~ I

10

PB

Fuzzy parameters description for the Anti-lock (ABS) control problem

r:1 n£. ('

~13

.~ . -,

-:-.

·3(1.:;0

_.,1.1

-1 ~

·1 .7

_ ~(It ) .

. ;.\1 11 1

--'~(I

_ ~ f ll l

'\"\1

.-

l\:S

,

LE

PS 1

~

11

P ~I

3

,

I'll

I :'

31)

-' -'(1

~ (M".I

('

6TB

I.'

::: 00

1\1)(1

11

Fig . 8. Fuzzy membership functions for the ABS control problem The fuzzy logic design can be verdied by applying thc proposed fuzzy logic controller above to the wheel speed dynamics of equations (3 . 1)-(34) The fuzzy logic representation of Table 2 can be used to describe I-ln.l T and subsequently F D in equation (3 . I) . The following initial state is assumed (!) (0) = 109 rad/sec and TB (0) = 22856 Ib-ft The profile of the II1put torque is shown in Figure 9 (see Fig. 9) . 391

fuzzy inference is another important issue that requires more study. x 10

4

2 .3:a------ - -- -- . . - - -I

5.

REFERENCES

The Highway Loss Data Institute ( 1994) Antilock brakes don't work as well as thought. San Jose Mercllry Ne . . . .- s, January 27 .

228:

7B 22& 2241

22~

Mamdani, E. H. and Assilian , S (1975) An experiment in linguistic synthesis \\'ith a fuzzy logic controller. Int. J Man-Machine ,S'llIdies, Vol 7 No L pp 751-769.

i

I

;

2i -- - ----- - - ·--- .-----~ o

1

234 llmc(s(.' (

5

I

Tseng , H.C. and Hwang, V H (1993) Seno controller tUlllng with fuzzy logiC IEEE

Fig . 9. Profile of final braking torque subject to FLC

Tral1S . on Conlrol S).srems and Technology.

Vol I. No 4_ pp .262-269

Figure 10 ShO\\S th:lt the slmul:lted III (t) cune subject to our fuzzy controllcr II1dced con\'crgcs to (,) in 2.5 scconds (see Fig 10) Tuning the fuzzy memberslllp functions C:ln bring thc settling time do\\'n to the order of 0 I seconds .

Z:ldeh. L. A (1965) Fuzzy sets . InformOl Comrol. Vol 8, pp 33 8-353 Z:ldeh , L. A (197 I) Similarity Relations and Fuzzy Ordenng. Informal Sciences. Vol 3, pp 177-200

109, -- ------- - - .

1085

(I)

107 5

106 .5 -- - ---- -- - -· - --· -o 1 2 3 4 5

Fig 10. Profile of final wheel speedsubject to FLC

4.

CONCLUSION

Fuzzy logic IS shown to be a successful tool in our simplified ABS problem. "here the aircraft speed is assumed constant. The uncertain and non linear features of the ABS design find fuzzy logic as an ideal selling to cope \\ith such Other aerospace and \'ehicle control issues. such as the adaptive suspension system. autonomous na\'igation, and engine control. to name.i ust a fe\\ . \\'i 11 also find the fuzzy logic approach useful Future research calls for the A BS design for the realistic system where the aircraft dynamics are coupled with the dynamics of several wheels. Blending the trainability of neural networks \\ith the adaptive 392